The analysis, propagation, and prediction of multivariate noise after a series of operations is a basic problem arising in many applications such as: audio signal processing, image denoising, digital imaging system design, seismic wave data analysis, and medical imaging. The signal processing steps that constitute these operations are able to be classified into three categories: linear, non-linear, and spatial. Existing methods are able to handle linear and non-linear transformations. However, they either ignore spatial signal processing steps, resulting in inaccurate noise propagation, or they employ convolution which is computationally very expensive. Therefore, they make the scheme impracticable for applications such as multi-parameter optimization, noise cancellation in audio, and predicting sub-band noise in real-time denoising.
If noise at the input of a spatial transformation is considered to be white, then the noise variance after filtering is able to simply be computed as the product of the input noise energy and the filter energy. Unfortunately, this assumption is not valid for most systems. Although the noise may be white at the beginning, its characteristics will change as it passes through a signal processing pipeline. Therefore, a mechanism is needed that is able to deal with both white and colored noise propagation.